An Iterative Spectral Approach to Recovering Planted Partitions

TL;DR: An efficient spectral algorithm is given that recovers the clusters with high probability, provided that the sizes of any two clusters are either very close or separated by sqrt(n).

Abstract: In the planted partition problem, the n vertices of a random graph are partitioned into k "clusters," and edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 < q < p < 1). In this work, we give an efficient spectral algorithm that recovers the clusters with high probability, provided that the sizes of any two clusters are either very close or separated by sqrt(n). The algorithm recovers the clusters one by one via iterated projection: it constructs the orthogonal projection operator onto the dominant k-dimensional eigenspace of the random graph's adjacency matrix, uses it to recover one of the clusters, then deletes it and recurses on the remaining vertices.